What is the state of a juggling sequence?

For a xed b, the state diagram Db for b balls is a directed graph with vertices being the juggling states with j j = b, and with directed edges de ned by the above transformations.

What is the result of the de nition of a juggling sequence?

It is an immediate consequence of the de nition of a juggling sequence that the \\number of balls" of a juggling sequence is always an integer (fortunately!).

Who is the author of the state diagram for juggling sequences?

(From a historical perspective, the notion of the state vector and the state diagram for juggling sequences appears to be due to Jack Boyce (see [10]).

What is the general definition of a juggling sequence?

In general, a juggling sequence T with state (i.e., which corresponds to a cycle in Db starting and ending at ) will be called a -juggling sequence.

Open AccessJournal Article10.1080/00029890.2008.11920516

Primitive Juggling Sequences

Fan Chung, +1 more

- 01 Mar 2008

- American Mathematical Monthly

- Vol. 115, Iss: 3, pp 185-194

TL;DR: (2008).

Abstract: (2008). Primitive Juggling Sequences. The American Mathematical Monthly: Vol. 115, No. 3, pp. 185-194.

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Most frequently asked questions

1. What are the contributions mentioned in the paper "Primitive juggling sequences" ?

In this note, the authors analyze the juggling sequences associated with an arbitrary state, and show how to enumerate the number of such sequences which have a given length.

2. What is the use of primitive juggling sequences?

As the authors will see, one use of primitive juggling sequences is that the authors can make new juggling sequences by taking arbitrary products of -primitive juggling sequences.

3. what is the recurrence relation for a matrix?

Suppose M is a binary matrix with rows and columns indexed by f1; 2; : : : ; ng such that the jth column of M consists of vj 1's above n vj 0's.

4. what is the recurrence relation for b?

For general b, the recurrence relation for P (b; n) can be derived from the generating function gb(x) in (4):P (b; n) = b 1X k=0 (b k)k!P (b; n k 1)which explains the coe cients in (3).

Figure 1: A juggling sequence and its associated state.

Figure 2: The state diagram for 3 balls and height 5.

Citations

•Journal Article•10.1112/S0010437X13007240

Positroid varieties: juggling and geometry

Allen Knutson, +2 more

- 01 Oct 2013

- Compositio Mathematica

TL;DR: In this paper, it was shown that the intersection of only the cyclic shifts of one Bruhat decomposition has many of the good properties of the Bruhat and Richardson decompositions.

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244

•Posted Content

Positroid varieties I: juggling and geometry

Allen Knutson, +2 more

- 21 Mar 2009

- arXiv: Algebraic Geometry

TL;DR: In this article, the authors show that the strata positroid varieties are normal, Cohen-Macaulay, have rational singularities, and are defined as schemes by the vanishing of Plucker coordinates.

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•Journal Article•10.1016/J.EJC.2010.01.011

Descent polynomials for permutations with bounded drop size

Fan Chung, +3 more

- 01 Oct 2010

- The Journal of Combinatorics

TL;DR: In this paper, the authors examined permutations on the set {1,2,...,n} with d descents and maximum drop size k. They gave explicit formulas for enumerating such permutations for given integers k and d.

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•Journal Article•10.1007/S00026-009-0040-Y

Enumerating (Multiplex) Juggling Sequences

Steve Butler, +1 more

- 12 Jan 2010

- Annals of Combinatorics

TL;DR: In this paper, the problem of enumerating periodic σ-juggling sequences of length n for multiplex juggling, where σ is the initial state (or landing schedule) of the balls, was studied.

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•Journal Article•10.46298/DMTCS.2856

Descent polynomials for permutations with bounded drop size

Fan Chung, +3 more

- 01 Jan 2010

- Discrete Mathematics & Theoretical Compu...

TL;DR: Permutations on the set {1,2,...,n} with d descents and maximum drop size k are examined, and explicit formulas for enumerating such permutations for given integers k and d are given.

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References

•Book

Enumerative Combinatorics

R P Stanley

- 13 Apr 1997

Abstract: Richard Stanley's two-volume basic introduction to enumerative combinatorics has become the standard guide to the topic for students and experts alike. This thoroughly revised second edition of volume two covers the composition of generating functions, in particular the exponential formula and the Lagrange inversion formula, labelled and unlabelled trees, algebraic, D-finite, and noncommutative generating functions, and symmetric functions. The chapter on symmetric functions provides the only available treatment of this subject suitable for an introductory graduate course and focusing on combinatorics, especially the Robinson–Schensted–Knuth algorithm. An appendix by Sergey Fomin covers some deeper aspects of symmetric functions, including jeu de taquin and the Littlewood–Richardson rule. The exercises in the book play a vital role in developing the material, and this second edition features over 400 exercises, including 159 new exercises on symmetric functions, all with solutions or references to solutions.

...read moreread less

7.8K

•Book

Concrete Mathematics: A Foundation for Computer Science

Ron Graham, +2 more

- 01 Jan 1994

TL;DR: This book introduces the mathematics that supports advanced computer programming and the analysis of algorithms, and is an indispensable text and reference not only for computer scientists - the authors themselves rely heavily on it!

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3.1K

Journal Article•10.2307/2324448

Concrete Mathematics: A Foundation for Computer Science.

Edward A. Bender, +3 more

- 01 Oct 1991

- American Mathematical Monthly

TL;DR: Concrete Mathematics as discussed by the authors is a collection of techniques for solving problems in computer science, and it is an indispensable text and reference not only for computer scientists - the authors themselves rely heavily on it! - but for serious users of mathematics in virtually every discipline.

...read moreread less

2.8K

Journal Article•10.1007/978-1-4615-9763-6

Enumerative Combinatorics

Richard P. Stanley

- 31 Jul 1986

2.2K

•Book

Handbook of Combinatorics

Ron Graham, +2 more

- 25 Dec 1995

TL;DR: The history of combinatorics can be traced back to the work of A.C. Biggs et al. as discussed by the authors, who introduced the concept of graph embeddings.

...read moreread less

1.7K

...

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Primitive Juggling Sequences

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Most frequently asked questions

1. What are the contributions mentioned in the paper "Primitive juggling sequences" ?

2. What is the use of primitive juggling sequences?

3. what is the recurrence relation for a matrix?

4. what is the recurrence relation for b?

5. What is the state of a juggling sequence?

6. What is the result of the de nition of a juggling sequence?

7. Who is the author of the state diagram for juggling sequences?

8. What is the general definition of a juggling sequence?

Figures

Citations

Positroid varieties: juggling and geometry

Positroid varieties I: juggling and geometry

Descent polynomials for permutations with bounded drop size

Enumerating (Multiplex) Juggling Sequences

Descent polynomials for permutations with bounded drop size

References

Enumerative Combinatorics

Concrete Mathematics: A Foundation for Computer Science

Concrete Mathematics: A Foundation for Computer Science.

Enumerative Combinatorics

Handbook of Combinatorics

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